Quasi-Nonexpansive Mappings

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Convergence and Stability of Modified Random SP-Iteration for Generalized Asymptotically

Quasi-Nonexpansive Mappings

Rashwan Ahmed Rashwan

^?

and Hasanen Abuelmagd Hammad

Abstract

The purpose of this paper is to study the convergence and the almost sure T−stability of the modified SP-type random iterative algorithm in a separable Banach space. The Bochner integrability of random fixed points of this kind of random operators, the convergence and the almost sureT−stability for this kind of generalized asymptotically quasi-nonexpansive random mappings are obtained. Our results are stochastic generalizations of the many deterministic results.

Keywords: Almost sureT−stability, separable Banach spaces, random modified SP-iteration, generalized asymptotically quasi-nonexpansive, random fixed point.

2010 Mathematics Subject Classification: 47H09,47H10,54H25,49M05.

1. Introduction

Random fixed point theory of single-valued and multivalued random operators are the stochastic generalization of the classical fixed point theory for single-valued and multivalued deterministic mappings. The study of random fixed point theory was initiated in 1950 by Prague school of probabilistic. Many authors are impressed by random fixed point theory especially, when Bharucha-Reid [5, 6] presented his papers and his result led to the development of these theorems. Some authors (see, [2, 20, 21, 25]) have shown that under some assumptions, the existence of a deterministic fixed point is equivalent to the existence of a random fixed point. in this case, every deterministic fixed point theorem produces a random fixed point

?Corresponding author (E-mail: rr_rashwan54@yahoo.com) Academic Editor: Sirous Moradi

Received 14 December 2016, Accepted 12 March 2017 DOI: 10.22052/mir.2017.70306.1050

c

2017 University of Kashan

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theorem. After papers of Bharucha-Reid [5, 6], Špaček [23] and Hanš [9] introduced stochastic extension of the Banach fixed point theorem in a separable metric space. Itoh [10] in 1979, generalized and extended Špaček and Hanš’s theorem to a multivalued contraction random operators. Several random fixed point theorems for measurable closed and nonclosed valued multifunctions satisfying general continuity conditions are proved by Papageorgiou [13]. His results improves the results of Engl [8], Itoh [10] and Reich [16].

In 1999, Shahzad and Latif [22] introduced a general random fixed point theorem for continuous random operators. As applications, they stated and proved a number of random fixed points theorems for various classes of 1-set and 1-ball contractive random operators. Rashwan and Hammad [15], proved random fixed point theorems with an application to a random nonlinear integral equation. Chang et al. [7], Beg and Abbas [1] proved some convergence theorems of random Ishikawa scheme and random Mann iterative scheme for strongly pseudo-contractive operators and contraction operators, respectively, in separable reflexive Banach spaces.

Recently, Zhang et al. [26] studied the almost sure T−stability and convergence of Ishikawa-type and Mann-type random algorithms for certainφ−weakly contractive type random operators in a separable Banach space. They established the Bochner integrability of random fixed points for this kind of random operators and the almost sureT−stability and convergence for these two kinds of random iterative algorithms under suitable conditions. Okeke and Eke [12], extended the results of Zhang et al. [26] by introducing a Noor-type random iterative scheme and studying the same results. These results are the stochastic generalization of the deterministic fixed point theorems of Berinde [3, 4] and Rhoades [17, 18].

In 2011, Phuengrattana and Suantai [14] introduced the following iteration scheme:

LetC be a nonempty subset of a Banach spaceX (1) SP-Iteration: Choosex1∈C and define







xn+1= (1−αn)yn+αnT yn

yn= (1−βn)zn+βnT zn

zn= (1−γn)xn+γnT xn

n≥1, (1.1)

where{αn},{βn}and{γn}are sequences in[0,1].

Saluja [19] modified the iteration scheme (1.1) for three generalized asymptotically quasi-nonexpansive self mappings ofC as follows:

(2) Modified SP-Iteration: Choosex1∈C and define







xn+1 = (1−αn)yn+αnT₁ⁿyn

yn= (1−βn)zn+βnT₂ⁿzn

zn = (1−γn)xn+γnT₃ⁿxn

n≥1, (1.2)

where{αn},{βn}and{γn}are sequences in[0,1].

Remark 1. If we putT₁ⁿ =T₂ⁿ =T₃ⁿ=T for alln≥1in (1.2) then we obtain the SP-iteration (1.1) for generalized asymptotically quasi-nonexpansive self mappings ofC.

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The aim of this study is to introduce the modified SP-type random iterative scheme and study the convergence and almost sureT−stability of our newly random iteration. Also we prove the Bochner integrability of random fixed points for this kind of generalized asymptotically quasi-nonexpansive random mappings in a separable Banach space.

2. Preliminaries

Let(Ω,Σ, µ)is a complete probability measure space where µbe a measure,Σbe aσ−algebra subset ofΩandCis a nonempty subset of a separable Banach space X.

Definition 2.1. [11] A random variable x(ω)is Bochner integrable if kx(ω)k ∈ L¹(Ω,Σ, µ)implies that

Z

Ω

kx(ω)kdµ(ω)<∞. (2.1)

Proposition 2.2. [11] A random variablex(ω)is Bochner integrable if and only if the sequence of random variables{xn(ω)}^∞_n=1 converges strongly tox(ω)almost surely such that

n→∞lim Z

Ω

kxn(ω)−x(ω)kdµ(ω) = 0. (2.2)

Definition 2.3. [26] Assume that (Ω,Σ, µ) is a complete probability measure space andCis a nonempty subset of a separable Banach spaceX. LetT : Ω×C→ C be a random operator and the set F(T) = {ζ(ω) ∈ C : T(ω, ζ(ω)) = ζ(ω), ω∈Ω}denotes the set of random fixed points ofT.For any given random variable x◦(ω)∈C, define an iterative scheme{xn(ω)}^∞_n=0⊂C by

x_n+1(ω) =f(T, x_n(ω)), n= 0,1,2, ... (2.3) wheref is a measurable function in the second variable.

LetT has a random fixed point (sayζ(ω))which is Bochner integrable with respect to{xn(ω)}^∞_n=0. Let {yn(ω)}^∞_n=0⊂C be an arbitrary sequence of a random variable. Assume that

εn(ω) =kyn+1(ω)−f(T, yn(ω))k, (2.4) and considerkεn(ω)k ∈L¹(Ω,Σ, µ)where n= 0,1,2, ..,then the iterative scheme (2.3) is stable with respect toT almost surely (T−stable almost surely) if and only if

n→∞lim Z

Ω

kε_n(ω)kdµ(ω) = 0, (2.5)

implies thatζ(ω)is Bochner integrable with respect to{yn(ω)}^∞_n=0.

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Definition 2.4. LetCbe a nonempty subset of a separable Banach spaceX and T : Ω×C→C be a random mapping. The random mappingT is said to be:

(i) Nonexpansive if

kT(ω, x)−T(ω, y)k ≤ kx(ω)−y(ω)k,

for eachω∈Ω,wherex(ω), y(ω) : Ω→C are some measurable mappings, (ii) Quasi-nonexpansive if

kT(ω, x)−p(ω)k ≤ kx(ω)−p(ω)k,

for each ω ∈ Ω, where p(ω) : Ω → C is a random fixed point of T and x(ω) : Ω→Cis any measurable map,

(iii) Asymptotically nonexpansive if there exists a sequence of measurable mappings{kn(ω)}: Ω→[1,∞)withlim_n→∞k_n(ω) = 0such that

kTⁿ(ω, x)−Tⁿ(ω, y)k ≤(1 +kn(ω))kx(ω)−y(ω)k,

for eachω∈Ω,wherex(ω), y(ω) : Ω→C are some measurable mappings,

(iv) Asymptotically quasi-nonexpansive if there exists a sequence of measurable mappings{kn(ω)}: Ω→[1,∞)withlim_n→∞kn(ω) = 0such that

kTⁿ(ω, x)−p(ω)k ≤(1 +kn(ω))kx(ω)−p(ω)k,

for each ω ∈ Ω, where p(ω) : Ω → C is a random fixed point of T and x(ω) : Ω→Cis any measurable map,

(v) Generalized asymptotically quasi-nonexpansive if there exists two sequences of measurable mappings{rn(ω)},{sn(ω)}: Ω→[1,∞)withlim_n→∞rn(ω) = lim_n→∞sn(ω) = 0such that

kTⁿ(ω, x)−p(ω)k ≤(1+rn(ω))kx(ω)−p(ω)k+sn(ω)kx(ω)−Tⁿ(ω, x(ω))k, for each ω ∈ Ω, where p(ω) : Ω → C is a random fixed point of T and x(ω) : Ω→Cis any measurable map.

We note that a nonexpansive mapping is quasi-nonexpansive but not conversely and a nonexpansive mapping is an asymptotically nonexpansive mapping, an asymptotically nonexpansive mapping is an asymptotically quasi-nonexpansive mapping and an asymptotically quasi-nonexpansive mapping is a generalized asymptotically quasi-nonexpansive mapping.

Inspired and motivated by [19] we introduce a stochastic analogue of the modify iteration scheme (1.2) for three generalized asymptotically quasi-nonexpansive self mappings ofC as follows:

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(3) Modified Random SP-Iteration:







x_n+1(ω) = (1−α_n)y_n(ω) +α_nT₁ⁿ(ω, y_n(ω)) yn(ω) = (1−βn)zn(ω) +βnT₂ⁿ(ω, zn(ω)) zn(ω) = (1−γn)xn(ω) +γnT₃ⁿ(ω, xn(ω))

n≥1, (2.1)

where{α_n},{β_n}and{γ_n}are real sequences in[0,1].

The following lemma will be needed in this study.

Lemma 2.5. [24] Let the sequences{an} and{un} of real number satisfy

an+1≤(1 +un)an,

wherea_n≥0, u_n ≥0, for alln= 1,2, ... and

∞

P

n=1

u_n <∞. Then

(i) lim

n→∞a_n exists,

(ii) if lim

n→∞infa_n= 0, then lim

n→∞a_n= 0.

3. The Convergence Results

Theorem 2.1. Let(X,k.k)be a separable Banach space andT1, T2, T3: Ω×C→C be three generalized asymptotically quasi-nonexpansive random mappings with two sequences of measurable mappings{rn(ω)},{sn(ω)}: Ω→[1,∞),lim_n→∞rn(ω) = lim_n→∞s_n(ω) = 0such that

∞

P

n=1

r_n(ω)+2s_n(ω)

1−sn(ω) <∞and

3

T

i=1

F(T_i)6=∅.Let{xn(ω)}

be a modified SP-random iterative sequence defined by (2.1) where{αn},{βn} and {γn} are real sequences in [0,1] and x^∗(ω) is a common random fixed point of T1, T2 andT3. Then the common random fixed point x^∗(ω)is Bochner integrable provided that lim

n→∞infR

Ω

kxn(ω)−x^∗(ω)kdµ(ω) = 0.

Proof. It suffices to show that

n→∞lim Z

Ω

Forω∈Ω,then from (2.1) we get

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Z

Ω

kx_n+1(ω)−x^∗(ω)kdµ(ω) ≤ (1−α_n) Z

Ω

ky_n(ω)−x^∗(ω)kdµ(ω)

+ αn

Z

Ω

kT₁ⁿ(ω, yn(ω))−x^∗(ω)kdµ(ω)

= (1−αn) Z

Ω

kyn(ω)−x^∗(ω)kdµ(ω)

+ αn





(1 +r_n(ω))R

Ω

kyn(ω)−x^∗(ω)kdµ(ω) +sn(ω)R

Ω

kyn(ω)−T₁ⁿ(ω, yn(ω))kdµ(ω)





= (1 +αnrn(ω)) Z

Ω

+ αnsn(ω) Z

Ω

kyn(ω)−T₁ⁿ(ω, yn(ω))kdµ(ω).

It’s obvious thatαnrn(ω)≤rn(ω)andαnsn(ω)≤sn(ω),hence Z

Ω

kxn+1(ω)−x^∗(ω)kdµ(ω)≤





(1 +rn(ω))R

Ω

kyn(ω)−x^∗(ω)kdµ(ω) +sn(ω)R

Ω

kyn(ω)−T₁ⁿ(ω, yn(ω))kdµ(ω)



. (3.1) Now, we have

Z

Ω

kyn(ω)−T₁ⁿ(ω, yn(ω))kdµ(ω) ≤ Z

Ω

+ Z

Ω

kT₁ⁿ(ω, y_n(ω))−x^∗(ω)kdµ(ω)

≤ Z

Ω

+ (1 +rn(ω)) Z

Ω

+ s_n(ω) Z

Ω

ky_n(ω)−T₁ⁿ(ω, y_n(ω))kdµ(ω)

= (2 +rn(ω)) Z

Ω

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+s_n(ω)R

Ω

kyn(ω)−T₁ⁿ(ω, y_n(ω))kdµ(ω), which implies that

Z

Ω

kyn(ω)−T₁ⁿ(ω, y_n(ω))kdµ(ω)≤

2 +r_n(ω) 1−sn(ω)

Z

Ω

kyn(ω)−x^∗(ω)kdµ(ω).

(3.2) Using (3.2) and (3.1), it follows that

Z

Ω

1 +r_n(ω) +s_n(ω) 1−sn(ω)

Z

Ω

(3.3) Again from (2.1), we can write

Z

Ω

kyn(ω)−x^∗(ω)kdµ(ω) ≤ (1−βn) Z

Ω

kzn(ω)−x^∗(ω)kdµ(ω)

+ β_n Z

Ω

kT₂ⁿ(ω, z_n(ω))−x^∗(ω)kdµ(ω)

≤ (1−βn) Z

Ω

+ βn





(1 +r_n(ω))R

Ω

kz_n(ω)−x^∗(ω)kdµ(ω) +sn(ω)R

Ω

kzn(ω)−T₂ⁿ(ω, zn(ω))kdµ(ω)





≤ (1 +rn) Z

Ω

+ sn(ω) Z

Ω

kzn(ω)−T₂ⁿ(ω, zn(ω))kdµ(ω). (3.4)

By the same manner of (3.2), we have Z

Ω

kzn(ω)−T₂ⁿ(ω, z_n(ω))kdµ(ω)≤

2 +r_n(ω) 1−sn(ω)

Z

Ω

kzn(ω)−x^∗(ω)kdµ(ω).

(3.5) Applying (3.5) in (3.4), we get

Z

Ω

kyn(ω)−x^∗(ω)kdµ(ω)≤

1 +r_n(ω) +s_n(ω) 1−sn(ω)

Z

Ω

(3.6)

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Substituting (3.6) in (3.3), we obtain that Z

Ω

1 +rn(ω) +sn(ω) 1−sn(ω)

²Z

Ω

(3.7) Similarly, one can easily write

Z

Ω

kzn(ω)−x^∗(ω)kdµ(ω)≤

1 +rn(ω) +sn(ω) 1−s_n(ω)

Z

Ω

kxn(ω)−x^∗(ω)kdµ(ω).

(3.8) From (3.8) and (3.7), it follows that

Z

Ω

kxn+1(ω)−x^∗(ω)kdµ(ω) ≤

1 +rn(ω) +sn(ω) 1−s_n(ω)

³Z

Ω

kxn(ω)−x^∗(ω)kdµ(ω)

=

1 +rn(ω) + 2sn(ω) 1−sn(ω)

³Z

Ω

= (1 +hn(ω))³ Z

Ω

≤ (1 +Qh_n(ω)) Z

Ω

kxn(ω)−x^∗(ω)kdµ(ω),

where hn(ω) = ^rⁿ^(ω)+2s_1−s ⁿ^(ω)

n(ω) for someQ > 0. Putting un =Qhn(ω) ≥0 in the above inequality and by

∞

P

n=1

rn(ω)+2s_n(ω)

1−sn(ω) < ∞, we have

∞

P

n=1

un < ∞. So, since

n→∞lim infR

Ω

kxn(ω)−x^∗(ω)kdµ(ω) = 0, all conditions of Lemma 2.1 are satisfied and

Z

Ω

This completes the proof.

4. The Stability Results

Theorem 2.1. Let(X,k.k)be a separable Banach space andT₁, T₂, T₃: Ω×C→C be three generalized asymptotically quasi-nonexpansive random mappings with two sequences of measurable mappings{r_n(ω)},{s_n(ω)}: Ω→[1,∞),lim_n→∞r_n(ω) = lim_n→∞sn(ω) = 0such that

∞

P

n=1

rn(ω)+2sn(ω)

1−sn(ω) <∞and

3

T

i=1

F(Ti)6=∅. Let x^∗(ω)

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be a random fixed point ofT₁, T₂andT₃and{xn(ω)} be a modified SP-random iterative sequence defined by (2.1) converging strongly tox^∗(ω)almost surely, where {α_n},{β_n}and{γ_n}are real sequences in[0,1]. Then{x_n(ω)}isT−stable almost surely provided that lim

n→∞infR

Ω

kx_n(ω)−x^∗(ω)kdµ(ω) = 0.

Proof. Let{yn(ω)}^∞_n=1 be any sequence of random variable inC and

kεnk=kyn+1(ω)−(1−αn)mn(ω)−αnT₁ⁿ(ω, mn(ω))k, (4.1) where mn(ω) = (1−βn)zn(ω) +βnT₂ⁿ(ω, zn(ω)) and zn(ω) = (1−γn)yn(ω)− γnT₃ⁿ(ω, yn(ω))andlimn→∞R

Ω

kεnkdµ(ω) = 0for everyω∈Ω.Now we prove that x^∗(ω)is Bochner integrable with respect to the sequence{yn(ω)}^∞_n=1. By (4.1)

Z

Ω

kyn+1(ω)−x^∗(ω)kdµ(ω) ≤ Z

Ω

kyn+1(ω)−(1−αn)mn(ω)

−αnT₁ⁿ(ω, mn(ω))kdµ(ω) + (1−αn)

Z

Ω

kmn(ω)−x^∗(ω)kdµ(ω)

+ α_n Z

Ω

kT₁ⁿ(ω, m_n(ω))−x^∗(ω)kdµ(ω)

≤ Z

Ω

kεnkdµ(ω)

+ (1−αn) Z

Ω

kmn(ω))−x^∗(ω)kdµ(ω)

+ α_n



(1 +r_n(ω)) Z

Ω

+ sn(ω) Z

Ω

kmn(ω)−T₁ⁿ(ω, mn(ω))kdµ(ω)



.

Again sinceαnrn(ω)≤rn(ω)andαnsn(ω)≤sn(ω),it follows that

Z

Ω

kyn+1(ω)−x^∗(ω)kdµ(ω)≤



 R

Ω

kεnkdµ(ω) + (1 +rn(ω))R

Ω

kmn(ω))−x^∗(ω)kdµ(ω) +sn(ω)R

Ω

mn(ω)−T₁ⁿ(ω, mn(ω)) dµ(ω)



. (4.2)

To estimateR

Ω

kmn(ω)−T₁ⁿ(ω, mn(ω))kdµ(ω),using (3.2), we get

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Z

Ω

kmn(ω)−T₁ⁿ(ω, mn(ω))kdµ(ω)≤

2 +rn(ω) 1−sn(ω)

Z

Ω

kmn(ω)−x^∗(ω)kdµ(ω).

(4.3) Substituting (4.3) in (4.2), we obtain

Z

Ω

kεnkdµ(ω)

+

1 +rn(ω) +sn(ω) 1−s_n(ω)

Z

Ω

(4.4) Again to estimateR

Ω

kmn(ω)−x^∗(ω)kdµ(ω), by (3.4), (3.5) and (3.6), we get Z

Ω

kmn(ω)−x^∗(ω)kdµ(ω)≤

1 +rn(ω) +sn(ω) 1−sn(ω)

Z

Ω

(4.5) Applying (4.5) in (4.4), we get

Z

Ω

kεnkdµ(ω)

+

1 +rn(ω) +sn(ω) 1−sn(ω)

²Z

Ω

(4.6) Also to estimateR

Ω

kmn(ω)−x^∗(ω)kdµ(ω),by (3.8), (4.6) we arrive at

Z

Ω

kεnkdµ(ω)

+

1 +rn(ω) +sn(ω) 1−sn(ω)

3Z

Ω

ky_n(ω)−x^∗(ω)kdµ(ω)

= Z

Ω

kε_nkdµ(ω)

+

1 +rn(ω) + 2sn(ω) 1−sn(ω)

3Z

Ω

= Z

Ω

kεnkdµ(ω) + (1 +hn)³ Z

Ω

≤ Z

Ω

kεnkdµ(ω) + (1 +Qhn) Z

Ω

kyn(ω)−x^∗(ω)kdµ(ω),

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where hn(ω) = ^rⁿ^(ω)+2s_1−s ⁿ^(ω)

n(ω) for some Q > 0. Putting un =Qhn(ω) ≥0 in the above inequality and by

∞

P

n=1

rn(ω)+2s_n(ω)

1−sn(ω) < ∞, we have

∞

P

n=1

un < ∞. So, since lim_n→∞R

Ω

kεnkdµ(ω) = 0 and lim

n→∞infR

Ω

kxn(ω)−x^∗(ω)kdµ(ω) = 0, hence all conditions of Lemma 2.1 are satisfied and

n→∞lim Z

Ω

kyn(ω)−x^∗(ω)kdµ(ω) = 0.

Conversely, ifx^∗(ω)is Bochner integrable with respect to the sequence{y_n(ω)}^∞_n=1, then from (4.1), (4.3), (4.4), (4.5) and (4.6), we get

Z

Ω

kεnkdµ(ω) = Z

Ω

kyn+1(ω)−(1−αn)mn(ω)−αnT1ⁿ(ω, mn(ω))kdµ(ω)

≤ Z

Ω

kyn+1(ω)−x^∗(ω)kdµ(ω) + (1−αn) Z

Ω

+ αn

Z

Ω

kT₁ⁿ(ω, mn(ω))−x^∗(ω)kdµ(ω)

≤ Z

Ω

+ αn(1 +kn(ω)) Z

Ω

≤ Z

Ω

+ αn(1 +kn(ω))

1 +rn(ω) + 2sn(ω) 1−sn(ω)

3Z

Ω

(4.7) Taking the limit asn→ ∞in (4.7), we get

n→∞lim Z

Ω

kεnkdµ(ω) = 0.

This shows that the modified random SP-iteration process{xn(ω)}^∞_n=1isT−stable almost surely and the conclusion of Theorem 4.2 is proved.

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Rashwan Ahmed Rashwan Department of Mathematics, Faculty of Science,

Assuit University, Assuit 71516, Egypt

E-mail: rr_rashwan54@yahoo.com Hasanen Abuelmagd Hammad Department of Mathematics, Faculty of Science,

Sohag University, Sohag 82524, Egypt

E-mail: h_elmagd89@yahoo.com